Berry’s paradox

We begin with calling a positive integercurious if it can be defined
in the English language using no more than 1234 words. Since there
are finitely many English words, we see that there are only finitely
many curious positive integers.

Define n0 to be: the least positive integer that is not
curious.

n0 has just been described in 8≤1234 words, therefore, it
is curious after all!

The paradox above is called Berry’s Paradox. Berry’s
Paradox suggests the advantage of separating the language used to formulate
mathematical statements or theory (the object language) from the language
used to discuss those statements or the theory (the metalanguage).

Berry’s Paradox can be avoided by the following reformulation:

1.

fix the object language, called 𝐄∗;

2.

declare 𝐄∗ to be different from our
metalanguage, which is English here;

3.

define a curious positive integer to be one which can be
described in 𝐄∗ using no more than 1234 words of the
language;

4.

define n0 to be the least positive integer that is not
curious.

In the reformulation, we have defined curious positive integers and
n0 in English, which is not 𝐄∗. Thus, we have no
basis to conclude that n0 is curious, hence no contradiction
arises.

Commonly, 𝐄∗ is the first order logic. However, it is
not often necessarily the case, and 𝐄∗ above could
have been English anyway. We only need to formally distinguish the
statements formulating the mathematics from the statements
discussing those formulations, i.e., declaring the two classes of
statements to be disjunct, perhaps by italicizing the former.
Nevertheless, such approach evidently involves more work and is
understandably hard to follow.